Related papers: Collision Course for Objects Moving on a Spherical…
Motion planning for robotic manipulators relies on precise knowledge of the environment in order to be able to define restricted areas and to take collision objects into account. To capture the workspace, point clouds of the environment are…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position…
In this paper, we discuss the mechanics and planning algorithms to slide an object on a horizontal planar surface via frictional patch contact made with its top surface. Here, we propose an asymmetric dual limit surface model to determine…
We consider the problem of a sphere rolling of a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both…
We study experimentally the collision between a sphere falling through a viscous fluid, and a solid plate below. It is known that there is a well-defined threshold Stokes number above which the sphere rebounds from such a collision. Our…
We study the collision between the cue and the ball in the game of billiards. After studying the collision process in detail, we write the (rotational) velocities of the ball and the cue after the collision. We also find the squirt angle of…
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…
The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this…
Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we…
When granular systems are modeled by frictionless hard spheres, particle-particle collisions are considered as instantaneous events. This implies that while the velocities change according to the collision rule, the positions of the…
A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which…
We establish that many fundamental concepts and techniques in quantum field theory and collider physics can be naturally understood and unified through a simple new geometric language. The idea is to equip the space of collider events with…
This paper presents formulae for calculation the solid angle of intersecting spherical caps, conical surfaces and polyhedral cones.
We give a new proof of the formula expressing the area of the triangle whose vertices are the projections of an arbitrary point in the plane onto the sides of a given triangle, in terms of the geometry of the given triangle and the location…
We present a mathematical model to predict pedestrian motion over a finite horizon, intended for use in collision avoidance algorithms for autonomous driving. The model is based on a road map structure, and assumes a rational pedestrian…
This paper concerns the study of the homotopy type of the ordered configuration space for manifolds with boundary and as an application we will study the collision free motion planning problem on manifolds with boundary.
The motion of spinning particles around compact objects, for example a rotating stellar object moving around a supermassive black hole, is described by differential equations that are, in general, non-integrable. In this work, we present a…
In this article we introduce a variational approach to collision avoidance of multiple agents evolving on a Riemannian manifold and derive necessary conditions for extremals. The problem consists of finding non-intersecting trajectories of…
We analyze the frictionless motion of a point-like particle that slides under gravity on an inverted conical surface. This motion is studied for arbitrary initial conditions and a general relation, valid within 13%, between the periods of…